Quintic Fields Ramifying Above {5, 19}

Discriminant class: −19

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19  2 3 7 11 13 17 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1. 1_{k} 8_{+} 3_{i}  0 -40 15 1^{5} 8_{+} 1  1^{3} 2_{+} 1  2 4 5 3 4 5 5 4
S_{5}  110 -288  1^{2} 1_{i} 19∗ 19∗ 2 4 32 3 22 4 4 5
2. 1_{k} 8_{+} 3_{i}  0 -40 15 1^{5} 8_{+} 1  1^{3} 2_{+} 1  32 32 22 3 4 22 5 32
S_{5} 5  -365 2657  1^{2} 1_{i} 19∗ 19∗ 4 4 2 3 22 4 4 5
3. 1_{k} 8_{+} 3_{i}  0 -50 -25 1^{5} 8_{+} 1  1^{3} 2_{+} 1  32 4 5 22 2 5 3 4
S_{5} 5  -125 -825  1^{2} 1_{i} 19∗ 19∗ 4 32 4 3 3 32 32 22

Discriminant class: 1

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19  2 3 7 11 13 17 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
4. 2_{−} 6_{+} 2_{−} 0 -5 -10 1^{5} 6_{+} 1  1^{2} 1_{i} 19∗ 19∗ 5 5 22 5  22 22 22
D_{5}  20 13  1^{2} 1_{i} 19∗ 19∗ 22 5 22 22 22 5 22 5
5. 2_{−} 8_{+} 2_{−} 0 35 -40 1^{5} 8_{+} 1  1^{2} 1_{i} 19∗ 19∗ 22 22 5 5 22 5 5 22
D_{5} 5 160 -232 1^{2} 1_{i} 19∗ 19∗ 22 22 22 5 22 22 5

Discriminant class: 5

∞ 5 19 a_{1}  a_{2} a_{3} Over p = 5  Over p = 19 2 3 7 11 13 17 23 29
G h  a_{4}  a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
6. 2_{−} 3_{−} 4_{+}  -1  8 6 1^{4} 3_{−} 5 5 1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5}  7  2  5 4  4 4 4 22 5
7. 2_{−} 5_{−} 4_{+}  0  0 0 1^{5} 5_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 5  0  -19  5 4 5 4 4 4 22 5
8. 0_{+} 9_{−} 2_{−}  0  -25 0 1^{5} 9_{−} 5  1^{2} 1_{k} 19 19 4 4 4 5 4 4 4 5
F_{5}  125  -25  1^{2} 1_{k} 19 19 22 4 22 4 4 4 5 5
9. 2_{−} 7_{−} 4_{+}  0  95 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 5  1805 -722  5 4 5 4 4 4 22
10. 2_{−} 7_{−} 4_{+}  0  -95 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 5  1805 -3249  4 5 4 4 4 22 5
11. 2_{−} 7_{−} 4_{+}  0  95 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4  4 4 4 22
F_{5} 5  1805  893  5 4 5 4 4 4 22 5
12. 2_{−} 7_{−} 4_{+}  0  -95 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 20  1805 -5529  5 4 5 4 4 4 22 5
13. 2_{−} 9_{−} 4_{+}  0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 10  0 -11875  5 4 5 4 4 4 22 5
14. 2_{−} 9_{−} 4_{+}  0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4  4 4 4 22
F_{5} 10  0 -1805  5 4  4 4 4 22 5
15. 2_{−} 9_{−} 4_{+}  0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 5  0  -95  5 4 5 4 4 4 22 5
16. 2_{−} 9_{−} 4_{+}  0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 5 4 4 4 22
F_{5} 10  0 -475  5 4 5 4 4 4 22

Discriminant class: 5 (continued)

∞ 5 19 a_{1}  a_{2} a_{3} Over p = 5  Over p = 19 2 3 7 11 13 17 23 29
G h  a_{4}  a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
17. 2_{−} 7_{−} 2_{+}  0  0 -15 1^{5} 7_{−} 5  1^{3} 2_{+} 1  4 2 32 3 32 4 4 5
S_{5}  65  -54  5 4 5 32 32 4 5 5
18. 2_{−} 5_{−} 4_{+}  0  0 -95 1^{5} 5_{−} 5  1^{5} 4_{+} 1  2 32 32 5 4 4 2 5
S_{5}  -190 -608  3 4 3 4 4 32 5 3
19. 2_{−} 9_{−} 2_{+}  0  25 -50 1^{5} 9_{−} 5  1^{3} 2_{+} 1  2 32 32 5 32 32 2 3
S_{5}  100  -80  5 4 3 4 32 4 22 22
20. 2_{−} 7_{−} 4_{+}  0  0 -95 1^{5} 7_{−} 5  1^{5} 4_{+} 1  32 2 4 3 32 32 4 5
S_{5} 5  -285 -513  3 2 22 32 32 2 22 5
21. 2_{−} 9_{−} 4_{+}  0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  32 4 4 3 2 32 32 5
S_{5}  1425  -95  22 32 22 32 4 32 3 5